What do people do these days for easily handling contact forms with a shared host? They come from my server name which then I change, so sometimes could be marked as spam. Simple php post. To/From
Always send those mails from a specifically configured real email address at your application's domain, not involving the name of your server. For instance, send the mail "From: contact-form@example.com", and "To: wherever-that-mail-goes@example.com". And send the mail using the actual mail server that handles @example.com mail (ideally not one you administrate). Don't include any user-supplied email address in either From or To; that'll get your mails either marked as spam or used as a vehicle for actual spam.
Mailgun.
Did this exist when Mandrill decided to boot their customers? I migrated all my clients that were on it to sendgrid but this would have been a good option for the cheap ones